Queuing theory

The primary objective of this thesis is to design and construct a program which will permit the user to obtain immediate results for a microcellular communication traffic engineering problem, although he may not have a thorough knowledge of telephone environment. The program is related to multi-channel, trunked communication systems, and it associates the number of channels and users as a function of system grade of service by implementing Poisson, Erlang B, Erland C and Crommelin-Pollaczek traffic models. The user will be able to choose between the different traffic models mentioned above via a menu. Then he will be asked to enter the parameters for the unknown variable(s) he wants to find. At this point, he will have to enter some other parameter values for which he is designing, and finally, the computer program will give the desired results (i.e., the number of servers needed, possible total traffic offered to the system, probability of loss or delay).

We consider the multiserver loss system with Poisson arrivals, and a different service-time distribution for each server. This system is a general case of the Erlang loss system, and hence is referred to as the general Erlang loss system (GELS). We show that when each server has the same mean service time the GELS with ordered entry, unlike the (ordinary) Erlang loss system with ordered entry, is sensitive to the service-time distributions. We also show that the order of selection of servers for the common-mean GELS with ordered entry has an effect on the loss probability of the system. With the aid of the software package MACSYMA, we obtain the exact expressions for the loss probabilities of three special cases of the 2-server GELS with ordered entry. For these special cases we also obtain the optimal order of selection of servers (that which gives least loss probability for the system). The issue of individual versus social optimization for the special cases is also considered here.

This thesis investigates the waiting-time in a token-ring local area
network (LAN) with gated service discipline. It gives an analytical
approximati.on for a network with an arbitrary number of stations,
non-identical Poisson arrival streams and general service time
distributions at each station, and non-zero switch-over time (also
referred to as walking time) between adjacent stations. This
approximation method is an alternative to the complicated O(N^2) numerical evaluation of the waiting-time currently available in the
literature. Extensive simulation results with different parameters are
presented to show the degree of accuracy of the approximation, which is
generally good for a practical range of parameters.

A polling model is a queueing model in which a single server visits N queues, traveling from queue to queue and "polling" each queue to ascertain whether customers are waiting to be served. When the server leaves a queue, it requires a switchover time to move to the next queue in the polling sequence; then, it requires a setup time to prepare to serve the customers in the polled queue. Two particular models are of interest. In the state-independent (SI) model, the server sets up whether or not there are customers waiting in the polled queue at the polling epoch, whereas in the state-dependent (SD) model, the server will not set up if the polled queue is empty. Recently, it was discovered that a decomposition phenomenon applies to switchover times in some polling models. It was proved for the cyclic exhaustive-service and gated-service polling models that the mean waiting time in the nonzero-switchover-time model can be decomposed into two terms, (i) the mean waiting time in a "corresponding" zero switchover-times model, and (ii) a term that depends essentially on only the sum of the mean switchover times per cycle. Another interesting phenomenon of polling models, discovered via numerical examples, is that reducing setup (or switchover) times in the SI model can, surprisingly, produce an increase in mean waiting times (Anomaly1, AN1). Equations were derived that explicitly characterize this anomaly. Also, numerical analysis of a 2-queue model showed that the SD model sometimes produces larger mean waiting times than the corresponding SI model (Anomaly 2, AN2). Explicit characterization of this anomaly for the symmetric case was done. In this dissertation we analyze and give extensions of both phenomena: decomposition and effects of setup time. Our contributions include: (1) We extend the decomposition theorem to include polling according to an order table (PAOT). (2) We generalize the decomposition theorem through providing a basic proof that applies to a variety of polling models. Models considered include (1) PAOT, (2) binomial service, (3) batch arrivals and (4) random polling. (3) We analyze the PAOT model using the buffer occupancy approach. This approach produces a set of equations that are applicable to our generalized proof of the decomposition theorem. (4) We prove that AN1 applies to the PAOT model, and use these results to obtain explicit formulas for the symmetric case that characterize this anomaly in the SI model. (5) We show via simulation that, under certain conditions, the anomalous result AN1, and AN2 uncovered for the SD model when N = 2 persist for models with N > 2.